What is the derivative of a(1-cos^2(x/2)^2) ?

The derivative of a(1-cos^2(x/2)^2) is 2asin^3(x/2)cos(x/2)

What is the first derivative of a(1-cos^2(x/2)^2) ?

The first derivative of a(1-cos^2(x/2)^2) is 2asin^3(x/2)cos(x/2)

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\frac{d}{d x} (a (1 - cos^{2} (\frac{x}{2}))^{2})

2 a sin^{3} (\frac{x}{2}) cos (\frac{x}{2})

Solution steps

\frac{d}{d x} (a (1 - cos^{2} (\frac{x}{2}))^{2})

Simplify

a (1 - cos^{2} (\frac{x}{2}))^{2} : a sin^{4} (\frac{x}{2})

= \frac{d}{d x} (a sin^{4} (\frac{x}{2}))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= a \frac{d}{d x} (sin^{4} (\frac{x}{2}))

Apply the chain rule:

4 (sin (\frac{x}{2}))^{3} \frac{d}{d x} (sin (\frac{x}{2}))

= 4 (sin (\frac{x}{2}))^{3} \frac{d}{d x} (sin (\frac{x}{2}))

\frac{d}{d x} (sin (\frac{x}{2})) = cos (\frac{x}{2}) \frac{1}{2}

= a \cdot 4 (sin (\frac{x}{2}))^{3} cos (\frac{x}{2}) \frac{1}{2}

Simplify

a \cdot 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) \frac{1}{2} : 2 a sin^{3} (\frac{x}{2}) cos (\frac{x}{2})

= 2 a sin^{3} (\frac{x}{2}) cos (\frac{x}{2})

What is the derivative of a(1-cos^2(x/2)^2) ?
The derivative of a(1-cos^2(x/2)^2) is 2asin^3(x/2)cos(x/2)
What is the first derivative of a(1-cos^2(x/2)^2) ?
The first derivative of a(1-cos^2(x/2)^2) is 2asin^3(x/2)cos(x/2)